All Questions
Tagged with automorphic-forms analytic-number-theory
114 questions
3
votes
0
answers
190
views
Voronoi formula on $\mathrm{GL}_4$ in the level aspect with ramification
$\DeclareMathOperator\GL{GL}$Let $f$ be an automorphic form on $\GL_3$ for $\Gamma_0(p)$ with $p$ being a prime (see Bump or Goldfeld's books for definitions). Recall that, in this paper-"The ...
- 41
8
votes
1
answer
356
views
Average bounds on Rankin-Selberg coefficients for modular forms
Let $f$ be a cuspidal Hecke newform of weight $k$ and level $N$, and denote by $a_f(n)$ its $n$-th Fourier coefficient. The newform $f$ is normalized so that $a_f(1) = 1$. As a consequence of Rankin-...
- 5,424
3
votes
1
answer
228
views
On the local factor of Rankin-Selberg L-functions
I have a puzzle on the local factors of Rankin-Selberg $L$-functions. Consider two newforms on $\text{GL}_2$. Let $f$ be a newform of square-free level $N$, and $g$ a newform of trivial level. As ...
- 353
1
vote
0
answers
68
views
Connection between a special integral transform and averages of L-functions
Let $\Gamma = \operatorname{SL}_2(\mathbb{Z})$ and $\mathcal{H}$ be the upper half-plane. For $A>1$, define the truncated Eisenstein series $E_A(z,s)$ as $$E_A(z,s) = \begin{cases} E(z,s), & \...
6
votes
1
answer
642
views
Generalizations of Hamburger's Theorem
(Despite the name, the theorem in question is not a joke nor is it a statement about a delicious food).
An old theorem of Hans Hamburger from 1921, as stated in Marvin Knopp's paper "On Dirichlet ...
- 26.9k
4
votes
1
answer
202
views
Is "self-dual" equivalent to "dihedral" for Maass forms on $\mathrm{GL}(2)$?
Suppose $f$ is a Maass cusp form on $\Gamma_0(D)$. The associated symmetric square $L$-function $L(\mathrm{sym}^2 f, s)$ has a pole at $s = 1$ if and only if $f = \overline{f}$ (if $f$ is self-dual).
...
- 1,570
3
votes
1
answer
178
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The lower bound for the automorphic $L$-function $L(s,\pi)$ at the edge of the critical strip $\Re s=1$
Let $\pi$ be any automorphic Maass form on $\text{GL}_m$ of level $N$, say. Assume that the associated $L$-function $L(s,\pi)$ satisfies some good conditions; for example, it satisfies the functional ...
- 81
5
votes
1
answer
162
views
A question on hybrid subconvexity for individual L-functions
Sorry to disturb. I have a question need some explanations from the experts on the MO-website.
As usual, we let $L(f,s)$ be the corresponding $L$-function associated to the newform $f$ on $SL_2(\...
- 81
3
votes
0
answers
78
views
shifted convolution in arithmetic progressions
Let $r(n)$ be the number of ways of writing $n$ as the sum of two integer squares. Asymptotics for the shifted convolution problem $$ \sum_{n\in \mathbb N\cap[1,x]}r(n) r(n+1)$$ are quite classical; a ...
- 3,062
5
votes
0
answers
133
views
Rankin-Selberg convolutions with mixed integral and half-integral weights
Let $f(z)$ denote a weight $0$ Hecke-Maass form of level $N$ and let $\theta(z)$ denote the Jacobi theta function. Then $y^{1/4} f(z) \overline{\theta(z)}$ transforms as an automorphic form of weight $...
- 51
3
votes
0
answers
122
views
Is there any notion of Poincaré series for Hermitian modular forms?
I have been studying modular forms and their generalisations for a year or so. It is a very interesting fact that the space of cusp forms $S_k$ is generated by the Poincaré series of exponential type (...
2
votes
0
answers
141
views
Analyticity of unramifed part of Rankin-Selberg $L$-functions on $\Re(s)=1$
I have only a little knowledge about automorphic representations and $L$-functions. Now I am reading the textbook of Goldfeld and Hundley on automorphic representations, and also planning to read the ...
- 663
2
votes
1
answer
264
views
'$\times$' or '$\otimes$' when writing $L$-functions?
Recently, I came across the Langlands correspondence theorem, there is the following line:
$$L(s,\pi(\sigma) \times \pi(\tau)) = L(s,\sigma \otimes \tau), $$
where $\sigma$ and $\tau$ are ...
- 185
2
votes
1
answer
147
views
On the square mean of Fourier coefficients of cusp forms
I have a question which may look naive for many experts here:
For any primitive holomorphic form $f$ of level $M$ ($M\in \mathbb{N}$), whether or not one has the lower bound that:
$$\sum_{X<n\le 2X}...
- 563
1
vote
0
answers
186
views
Consult a question about subconvexity bounds for symmetric-square L-functions in an Arxiv-eprint due to P. D. Nelson
Sorry to disturb, the experts here. Recently, I read a paper of Nelson ("Subconvex equidistribution of cusp forms: reduction to Eisenstein observables--"https://arxiv.org/pdf/1702.02908.pdf)....
- 563
4
votes
1
answer
299
views
The Wilton-type bounds involving half-integral weight cusp forms
There is a basic question which puzzles me for a while, and maybe look naive for some experts here. The question is the following:
Let $f(z)=\sum_{n\ge 1} a_f(n) n^{k/2-1/4}e(nz)\in S_{k+1/2}(4N)$ be ...
- 563
3
votes
1
answer
237
views
Experiments with Voronoï summation
In order to test my understanding of the Voronoï summation formula, I tried to apply it to a simple estimation of partial sums of Fourier coefficients of cusp forms. The result I obtained cannot ...
- 545
2
votes
0
answers
109
views
Whittaker function oscillation on diagonal near 0
In Proposition 1 of Blomer - Applications of the Kuznetsov formula on GL(3), bounds are given for the Whittaker function
$$W_{\nu_1, \nu_2}(y_1, y_2) = \mathcal W_{\nu_1, \nu_2}\begin{pmatrix} y_1y_2 &...
- 1,890
6
votes
1
answer
323
views
The error term for the second moment of Fourier coefficients of cusp forms with the level explicitly determined
There is a basis question which puzzles me for a while. The question is the following:
Let $p$ be a prime and $X\ge 2$. Let $f$ be a $GL_2$-newform of level $p$ and non-trivial
nebentypus, with the $n$...
- 563
1
vote
0
answers
87
views
what is the relationship betwen $L(s,sym^mf\times sym^mg)$ symmetric L function of $f$ and $g$ and $\lambda_{f}(n^m)$, $\lambda_{g}(n^m)$?
what is the relationship betwen $L(s,sym^mf\times sym^mg)$ symmetric L function of $f$ and $g$ and $\lambda_{f}(n^m)$, $\lambda_{g}(n^m)$ ?
- 11
8
votes
1
answer
245
views
Spectral decomposition of $\Gamma\backslash X$
Let $X$ be a reasonable manifold of non-positive curvature (could be $\mathbb{H}^n$, symmetric or locally symmetric space, homogeneous Hadamard manifold etc.), and let $\Gamma$ be a reasonable group ...
- 81
2
votes
1
answer
219
views
Voronoï summation for cusp forms with characters
In an attempt to solve an unrelated problem, I was led to the task of estimating/bounding from above sums of the form
$$\sum_{m=1}^\infty\lambda(m)e\left(-\frac{am}{q}\right)h(m)$$
where $\sum_{m=1}^\...
- 545
2
votes
0
answers
98
views
Extrema of real analytic Eisenstein series and more general modular functions
The real analytic Eisenstein series defined by the Poincare sum
$$E(s,z)=\sum_{(m,n)\neq (0,0)} {y^s\over |mz+n|^{2s}}$$
for $z\in{\mathbb H}$ and ${\rm Re}(s)>1$ is a manifestly $SL(2,{\mathbb Z})$...
- 21
4
votes
1
answer
291
views
The Langlands parameters of the symmetric cube lifts of cusp forms
I have a question concerning the Langlands parameters of the symmetric cube lifts. Lets $f$ be a $GL_2$-cusp form, and $\operatorname{sym}^2f$ the symmetric-square lift of it. Assume that $\alpha_1,\...
- 563
4
votes
0
answers
204
views
A question on the twisted symmetric square L-functions
Sorry to disturb. I have a puzzle which might be naive for many experts here.
Let $f$ be a Hecke newform of prime level $N$ on $\mathrm{GL}_2$, and $
\chi$ a primitive character of square-free ...
- 353
4
votes
1
answer
197
views
On the precise form of $\mathrm{GL}(3)$ (and others) L-functions
$\DeclareMathOperator\GL{GL}$Typical $\GL(3)$ automorphic L-functions are often depicted as looking like the following Dirichlet series:
$$L(s, \pi) = \sum_{n >0} \frac{A(1,n)}{n^s}$$
Is there a ...
- 415
1
vote
0
answers
109
views
Whether or not the Maass form for $\Gamma _0(N)$ on $GL(3)$ covers the classical symmetric lift of a newform on $GL(2)$?
I have a blur which needs some help from the experts here, and may look naive for some experts. Recently I read Zhou's paper "The Voronoi formula on $GL{(3)}$ with ramification" (https://...
- 563
10
votes
1
answer
440
views
A question on the period integral of Rankin-Selberg $L$-function
$\DeclareMathOperator\GL{GL}$Let $\Pi$ and $\pi$ be irreducible automorphic representations of $\GL_{n+1}(\mathbb{A}_F)$ and $\GL_n(\mathbb{A}_F)$ respectively, where $n \geq 2$, $F$ is a number field ...
- 421
4
votes
0
answers
135
views
Values at 1 of symmetric power L-functions of Maass cusp forms
I have a blur that whether one has $L(1,\text{sym}^2f)\ll \log^A q$ for some $A>0$? Here $f$ is assumed to be a Maass cusp form of square-free level $q$. If any experts here know something about ...
- 563
1
vote
1
answer
323
views
How to relate Rankin triple L-function to its Dirichlet series
I have a very tricky question which may look naive to many experts here.
Let $f$ be a newform of level prime $P$, and $g,h$ two newforms of level 1, respectively. These three forms $f,g,h$ are all of ...
- 563
6
votes
2
answers
392
views
A lower-bound for the square-mean of Fourier coefficients of cusp forms at primes argument
There is a basis question which puzzles me for a while. The question is the following:
Let $X\ge 2,$ and $\lambda(n)$ be the $n$-th Fourier coefficient of a $GL(2)$ newform of prime level $N>1$, ...
- 563
1
vote
1
answer
329
views
Behaviour of a certain $L$ function at $s=1$
I was going through this paper. Corollary 7.3.4 says the $L$-function $L(s,\pi, \rm{sym}^4)$ is holomorphic except possibly at $s=0,1$ and gives a necessary and sufficient condition for it to have a ...
- 424
4
votes
1
answer
223
views
Fourier expansion of $\mathrm{GL}(n)$ automorphic forms for congruence subgroups
$\DeclareMathOperator\GL{GL}$There are several books describing the classical Fourier/Whittaker expansion of automorphic forms on $\GL_3(\mathbb{R})$, e.g. Bump - Automorphic forms on $\GL_3(\mathbb{R}...
- 767
4
votes
1
answer
318
views
Watson's triple product for automorphic forms shifted by Maass rising operators
Let $\phi_i$ be a holomorphic Hecke eigencusp form of weight $k_i$ for $\Gamma = \mathrm{SL}_2(\mathbb{Z})$, or a Maass cusp form (we then say that $k_i=0$). We assume they are normalised such that $\...
- 153
1
vote
1
answer
249
views
What's the motivation for the $3$ appearing in Iwaniec and Kowalski's definition of the analytic conductor?
In their book Analytic Number Theory, Iwaniec and Kowalski, on page 95, define the analytic conductor by the following formula:
$\displaystyle{{\frak{q}}_{\infty}(s)=\prod_{j=1}^{d}\left(\vert s+\...
- 6,974
14
votes
1
answer
532
views
Bound for $GL(3)$ symmetric square
Let $\pi$ be an automorphic representation of $GL(3)$ over a number field. Let $a_n$ be the coefficients of $L(s, \pi, \mathrm{sym}^2)$. Do we know if
$$\sum_{n>0} \frac{|a_n|}{n^s}$$
and
$$\sum_{n&...
- 5,424
3
votes
0
answers
216
views
Maass--Selberg for any Eisenstein series on higher rank
Does there exist a Maass--Selberg relation for any Langlands Eisensein series on $\mathrm{GL}(n)$? By any I mean an Eisenstein series which is induced from any standard parabolic with any discrete ...
- 1,692
3
votes
1
answer
1k
views
Are there infinitely many L-rigs?
$\DeclareMathOperator{\Q}{\mathbb{Q}}$Call "L-rig" any class $\mathcal{L}$ of L-functions of automorphic representations of $\operatorname{GL}_{n}(\mathbb{A}_{\Q})$ for some $n$ belonging to ...
- 6,974
8
votes
2
answers
744
views
A question related to Hilbert modular form
This is a question related to Hilbert modular forms.
Let $\mathbb{K}=\mathbb{Q}(\sqrt D)$ be an imaginary quadratic field with discriminant $D<0$ and $\zeta (\text{mod } m)$ a Hecke character such ...
- 424
8
votes
1
answer
476
views
Equivalence between Ramanujan and Selberg conjectures
At first the Ramanujan conjecture for automorphic forms and the Selberg conjecture appear to be understood as totally independent. However, they are now known to be tighyly connected once viewed in ...
- 5,424
1
vote
0
answers
65
views
Meaning of extended principal part of weakly holomorhpic modular forms
In p.312 of 'Rhoades, Robert C., Linear relations among Poincaré series via harmonic weak Maass forms. Ramanujan J. 29 (2012), no. 1-3, 311–320', the author defines the extended principal part at ...
- 663
7
votes
1
answer
315
views
Intuition about how Voronoi formulas change lengths of sums
In reading the literature one encounters countless examples of Voronoi formulas, i.e., formulas that take a sum over Fourier coefficients, twisted by some character, and controlled by some suitable ...
- 198
0
votes
1
answer
410
views
Rankin-Selberg convolution and product of degrees as of Christmas 2019
Almost 5 years ago (time flies), I asked in Rankin-Selberg convolution and product of degrees whether the Rankin-Selberg convolution of two automorphic representations of respectively $\operatorname{...
- 6,974
8
votes
1
answer
595
views
Does the symmetric square L-function vanish at one?
Take a cuspidal automorphic representation $\pi$ of $GL(3)$ over a number field. My question is quite straightforward and can be related to this one :
Can $L(1, \pi, \mathrm{sym}^2)$ be zero? If ...
- 927
5
votes
0
answers
111
views
Archimedean L-factors for symplectic group
Let $\pi$ be an automorphic representation of $GSp(4)$. Provided a representation $r$ of the Langlands dual group of $GSp(4)$ (namely, the standard or the spinor one), it is possible to define a ...
- 927
6
votes
1
answer
361
views
Decay of matrix coefficients of non-tempered representation
A theorem of Cowling--Haagerup--Howe gives an effective decay rate of the matrix coefficients of a tempered representation $\pi$ of a semi-simple algebraic $G$ in terms of Harish-Chandra $\Xi$ ...
- 1,692
3
votes
0
answers
63
views
What is meant by singular hyperplane of $c(w, \cdot)$? (global intertwining operator related to Eisenstein series)
Let $P_0$ be a minimal $\mathbb{Q}$-parabolic subgroup of $G$, a semisimple linear algebraic group over $\mathbb{Q}$. Then $P_0 = M_0 N_0$ where $M_0$ is a Levi subgroup of $P_0$. Let $E^G_{P_0}$ be ...
- 3,625
5
votes
1
answer
616
views
Question on an application of Langlands' result on the constant term of Eisenstein series (Is this a typo?)
I would like to understand an argument in https://link.springer.com/content/pdf/10.1007/BF01393904.pdf, which uses Langlands' result on the constant term of Eisenstein series, but I'm not getting it ...
- 3,625
2
votes
0
answers
196
views
Trying to understand why Eisenstein series is well defined
I am struggling to see why Eisenstein series is well defined, and I would greatly appreciate clarification.
Let
$$
E(x, \lambda) = \sum_{\delta \in P(\mathbb{Q}) \backslash G(\mathbb{Q}) }
e^{\...
- 3,625
8
votes
1
answer
530
views
How strong is the requirement of being a Gelbart-Jacquet lift?
Let $\pi$ be a cuspidal automorphic representation of $\mathrm{GL}(3)$ over a number field $F$. I am wondering how general are Gelbart-Jacquet lifts of automorphic representations of $\mathrm{GL}(2)$ ...
- 5,424